Half-Life Calculator
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Results
Remaining amount (N): 25 g
| Half-lives | Time | Remaining | Percent |
|---|---|---|---|
| 0 | 0 y | 100 g | 100% |
| 1 | 5 y | 50 g | 50% |
| 2 | 10 y | 25 g | 25% |
| 3 | 15 y | 12.5 g | 12.5% |
| 4 | 20 y | 6.25 g | 6.25% |
| 5 | 25 y | 3.125 g | 3.125% |
| 6 | 30 y | 1.563 g | 1.563% |
| 7 | 35 y | 0.7813 g | 0.781% |
| 8 | 40 y | 0.3906 g | 0.391% |
| 9 | 45 y | 0.1953 g | 0.195% |
| 10 | 50 y | 0.09766 g | 0.098% |
Decay is random for each nucleus; this formula gives the expected amount for a large number of atoms. Times use a year of 365.25 days.
How to use the half-life calculator
- Choose the unknown: remaining amount, initial amount, time, or half-life.
- Pick an isotope to fill in its half-life, or enter your own half-life and unit. Any exponentially decaying quantity works, including drug levels or a cooling-off model.
- Enter the amounts in any unit (grams, percent, becquerels, counts per minute). Both amounts use the same unit, so only their ratio matters.
- Enter the time and its unit. The results include the decay constant, the mean lifetime and a schedule for ten half-lives.
Half-life formulas
λ = ln 2 ÷ t½ · τ = 1 ÷ λ = t½ ÷ ln 2
t = t½ × log₂(N₀ ÷ N)
t½ = t × ln 2 ÷ ln(N₀ ÷ N)
N₀ is the starting amount, N the amount left after time t, and t½ the half-life: the time for half of the remaining nuclei to decay. The decay constant λ is the probability per unit time that a given nucleus decays, and the mean lifetime τ is the average time a nucleus survives, about 1.443 half-lives. Because every half-life removes half of what is left, after 10 half-lives only 0.098% remains.
Worked examples
- Iodine-131: with a half-life of 8.0252 days, 100 g left for 30 days decays to 100 × (1/2)30 ÷ 8.0252 = 7.49 g.
- Radiocarbon dating: a sample with 25% of its original carbon-14 is two half-lives old: 2 × 5,700 = 11,400 years. After 1,000 years, 88.55% of the carbon-14 remains.
- Finding a half-life: if a sample's count rate falls from 1,000 to 250 counts per minute in 16 days, t½ = 16 × ln 2 ÷ ln 4 = 8 days, close to iodine-131.
Half-lives of common isotopes
Half-lives are evaluated nuclear data (ENSDF) from the IAEA Nuclear Data Section's LiveChart of Nuclides, the same evaluations distributed by the US National Nuclear Data Center. The last two columns are computed with the calculator's engine.
| Isotope | Symbol | Half-life | Decay | Known for | Left after 1 year | Time to fall to 1% |
|---|---|---|---|---|---|---|
| Fluorine-18 | ¹⁸F | 109.77 minutes | β⁺ / electron capture | PET scans | < 0.000001% | 12.2 h |
| Technetium-99m | ⁹⁹ᵐTc | 6.0072 hours | Isomeric transition | Medical imaging | < 0.000001% | 1.66 days |
| Radon-222 | ²²²Rn | 3.8222 days | α | Indoor radon gas | < 0.000001% | 25.4 days |
| Iodine-131 | ¹³¹I | 8.0252 days | β⁻ | Thyroid treatment; fission product | < 0.000001% | 53.3 days |
| Phosphorus-32 | ³²P | 14.266 days | β⁻ | Biology labeling | 1.96 × 10⁻⁶% | 94.8 days |
| Iodine-125 | ¹²⁵I | 59.407 days | Electron capture | Brachytherapy, lab assays | 1.41% | 1.08 years |
| Polonium-210 | ²¹⁰Po | 138.376 days | α | Static eliminators | 16% | 2.52 years |
| Cobalt-60 | ⁶⁰Co | 1925.28 days (5.27 years) | β⁻ | Radiotherapy, sterilization | 87.7% | 35 years |
| Tritium (hydrogen-3) | ³H | 12.32 years | β⁻ | Exit signs, watch dials | 94.5% | 81.9 years |
| Strontium-90 | ⁹⁰Sr | 28.91 years | β⁻ | Fission product | 97.6% | 192 years |
| Cesium-137 | ¹³⁷Cs | 30.08 years | β⁻ | Fission product, gauges | 97.7% | 200 years |
| Americium-241 | ²⁴¹Am | 432.6 years | α | Smoke detectors | 99.84% | 2,870 years |
| Radium-226 | ²²⁶Ra | 1,600 years | α | Historic luminous paint | 99.957% | 10,600 years |
| Carbon-14 | ¹⁴C | 5,700 years | β⁻ | Radiocarbon dating | 99.988% | 37,900 years |
| Plutonium-239 | ²³⁹Pu | 24,110 years | α | Nuclear fuel and weapons | 99.997% | 160,000 years |
| Uranium-235 | ²³⁵U | 704 million years | α | Nuclear fuel | > 99.999% | 4.68 × 10⁹ years |
| Potassium-40 | ⁴⁰K | 1.248 billion years | β⁻ / electron capture | Natural radioactivity, K–Ar dating | > 99.999% | 8.29 × 10⁹ years |
| Uranium-238 | ²³⁸U | 4.468 billion years | α | Uranium–lead dating | > 99.999% | 2.97 × 10¹⁰ years |
| Thorium-232 | ²³²Th | 14.0 billion years | α | Natural radioactivity | > 99.999% | 9.3 × 10¹⁰ years |
Where half-lives matter
Medicine favors short half-lives: technetium-99m loses 94% of its activity in a day, long enough for a scan but short enough to limit the patient's dose. Dating needs half-lives on the scale of the ages being measured: after 50,000 years less than 0.3% of the original carbon-14 is left, which is why radiocarbon dating stops working around that age and geologists turn to uranium or potassium isotopes. Waste management follows the same arithmetic: a store of cesium-137 needs about 300 years to fall to 0.1% of its original activity.
Things to know
- Decay is statistical. No one can say when a single nucleus will decay; the formula describes the average behavior of very many nuclei, which is extremely precise for a gram of material.
- Activity follows the same curve. Activity in becquerels or curies is proportional to the number of nuclei, so it halves every half-life too.
- Decay chains. Some products are themselves radioactive (radium-226 decays to radon-222, for example), so total radioactivity can behave differently from one isotope's curve.
- Seven half-lives leave under 1% (0.78%), a common rule of thumb for when a short-lived medical isotope has effectively gone.
- This calculator is for learning and planning. Radiation protection decisions should follow your radiation safety officer and official guidance.
The log calculator helps with the logarithms in the time and half-life formulas, and the exponent calculator with powers of ½.
Frequently asked questions
How do you calculate the amount left after a number of half-lives?
Use N = N₀ × (1/2)^(t/t½). 100 g of iodine-131 (half-life 8.0252 days) left for 30 days is 3.74 half-lives, so about 7.49 g remains.
How do I calculate a half-life from measurements?
Use t½ = t × ln 2 ÷ ln(N₀ ÷ N). If a count rate falls from 1,000 to 250 counts per minute in 16 days, that is two halvings, so t½ = 16 × 0.6931 ÷ 1.3863 = 8 days.
How old is a sample with 25% of its carbon-14 left?
Two half-lives. With the evaluated carbon-14 half-life of 5,700 years, that is 11,400 years. Real radiocarbon ages also need calibration for past changes in atmospheric carbon-14.
What is the decay constant?
The decay constant λ = ln 2 ÷ t½ is the fraction of nuclei that decay per unit time. For cesium-137 (half-life 30.08 years) it is about 0.02304 per year. The mean lifetime is its inverse, 1 ÷ λ, or about 1.443 half-lives.
How many half-lives until a substance is gone?
It never reaches exactly zero, but 7 half-lives leave under 1% (0.78%) and 10 half-lives leave about 0.098%. For cesium-137, falling to 1% takes about 200 years.